So...let's say we have an r-algebra. That's an algebra with an
invertible linear map
, which we draw
as a right-crossing:

which satisfies 1) the Yang-Baxter equations:

2)
and
,

and 3) the quasitriangularity conditions; writing multiplication as the
joining of strands these are

and

We say an r-algebra is ``r-commutative'' if

and ``strong'' if is the identity, or equals its
inverse, i.e.:

(Note that a left-crossing

is the inverse of a right-crossing.)

All sorts of noncommutative analogs of manifolds are r-commutative
algebras: quantum groups, noncommutative tori, quantum vector spaces,
the Weyl and Clifford algebras, certain universal enveloping algebras,
supermanifolds, etc. It seems that the ones with direct relevance to
quantum theory in 4 dimensions are ``strong,'' while the non-strong ones,
like quantum groups, are primarily relevant to 2- and 3-dimensional
physics. I would now like to describe an analog of differential forms
for strong r-commutative algebras, and illustrate it for the case of the
Heisenberg algebra -- i.e., the algebra defined by the canonical commutation
relations:
.

What are differential forms? Of course, they're the basis of a lot of
differential geometry, and there are lots of equivalent ways of defining
them, but let me take an algebraic viewpoint. Let be the algebra of
smooth functions on a manifold . We define differential forms as
follows. Each function in has a ``differential'' , and the
functions and their differentials generate an algebra in which we impose
the following relations:

Linearity:
, for any scalar .

The product rule:
.

The derivative of a constant vanishes: .

Commutation relations: and
.

This algebra is called the algebra of differential forms on .

That's all, folks! If you've taken a course in differential geometry
you were probably exposed to tangent planes and all that stuff, but if
you want to get calculating with differential forms as soon as possible
this is all you need to start with.

Note that rule 4) is the only one in which we switch elements of --
moving to the right of . This is the rule we'll need to modify for
an r-commutative algebra. You could, of course, just leave out rule 4):
given any algebra , the algebra whose relations are just given by 1)-3)
is called the ``universal differential calculus'' for . It's a
reasonable substitute for differential forms when is any old
noncommutative algebra, and (from one viewpoint) it's the basis of Alain
Connes' approach to noncommutative differential geometry. But when you
have a strong r-commutative algebra one can replace rule 3) with

4) Commutation relations:
and
.

Here I should explain that I'm writing as the sum over of
tensor products of the form
(one can always do this), and
I'm using the Einstein summation convention (sum over repeated indices)
to avoid writing the summation sign. What we're doing in rule 4) is
just what we should do: use to ``switch'' and instead of ``naively''
switching them as in 4).

Now let me show what this buys you in the case of the Heisenberg algebra.
This is the algebra over generated by 3 formal variables, , , and
, subject to the relations that

Note that we're not treating as a number here, but as a
variable, so I have to explicitly say that it commutes with everything.
(If were a number, it would be .)

This algebra is actually a strong r-commutative algebra in a unique
manner such that

and

Note what these say: the first two say that you switch with
and in the usual way, and the third one says that when you switch
and , you get the expected term and then a piece, , which comes from the fact that and
don't commute. The point is that while the Heisenberg algebra is not
commutative, the canonical commutations relations do tell you exactly what
to do when you switch and , which is just as good!

If I may digress...I happen to have the paper by Heisenberg, Born, and
Jordan with me, ``Zur Quantenmechanik II'', published in 1926, in which
the canonical commutation relations are introduced. I quote:

`Due to the failure of the commutative law for
multiplication, computations with quantum-theoretical quantities are
ambiguous in a certain sense, unless one prescribes a value for .
We introduce accordingly the following as the fundamental
quantum-mechanical relation:

Ain't that cool?
Anyway, we take the Heisenberg algebra as a strong r-commutative algebra and
define the differential forms on this algebra by the rules 1), 2) and
3'), we get the following standard-looking relations:

and a bunch of similar ones involving two differentials:

but then some more interesting ones:

It's a nice little exercise to see that these really do follow from
1)-4). One can have fun doing various things with these ``quantized
differential forms'' (and their generalizations), basically by taking your
favorite facts from the differential geometry of phase space and trying to
``quantize'' them, but let me just briefly run through the basics. (Now I
will let myself be more mathematical.) One may form a quotient of the
Heisenberg algebra by specializing the variable to some value; this is
called the Weyl algebra. The corresponding differential forms on the Weyl
algebra were discovered by I. Segal some time ago and called ``quantized
differential forms;'' their cohomology gives a nice way of understanding
the Wick product in quantum field theory. (He also dealt with a fermionic
version using the Clifford algebra.) In my paper on r-commutative geometry
I recommend an approach in which one views the Heisenberg algebra as a
``bundle'' over the ``line'' whose coordinate is given by . A given
``fiber'' (at which has some numerical value) is then a Weyl algebra
(except for the ``classical fiber'' at , which is just the algebra of
polynomials on phase space). I put quotes around the words ``fiber
bundle'' because only the classical fiber is really a manifold; the rest
are ``quantum manifolds,'' i.e. noncommutative algebras. But the fiber
bundle viewpoint gives good insight into the relation between the
differential forms on the base (the line), the fibers (Weyl algebras) and
the total space (the Heisenberg algebra). This viewpoint also works for
noncommutative tori (and other cases). Neither of these examples are
exciting from the point of view of braid invariants, since they are
``strong''. The differential forms as defined above do not work very well
for non-strong r-algebras. I have no idea whether there is a good general
definition of differential forms for r-commutative algebras; there is a
definition that works for the quantum plane and other quantum vector spaces
with Hecke-type relations (i.e., not , but
.) I
just finished writing a paper
in which I define a generalization of Hochschild homology for
r-algebras and relate it to the r-commutative differential forms defined
above. This paper is available here: